I’m not sure if this has already been brought up (it’s been a long thread), but I think that the presumption that you’re taking each term in a series one at a time (.9, .99, .999 etc.) is implicitly taking it as countable, when we know that the reals are uncountable.
The reals are uncountable, but that series is countable.
This thread is moving pretty fast, but I haven’t seen this referenced in the recent discussion about Zeno’s Paradox.
It seems that whenever someone gets stuck on Zeno’s, it’s assumed that it takes the same amount of time to go from 0.5 to 0.75 as it did to go from 0 to 0.5. It didn’t, it took half the time.
Sure, you have to move through an infinite number of halfway points from A to B, but the time it takes you to travel between those halfway points also decreases. I’m not sure why limits were off-limits in the explanation, because that’s where the paradox falls apart. As your distance from point B decreases to an infinitesimal number, so too does the time it take decrease to an infinitesimal number, until finally you’ve covered .9999… of the distance and have .0000… left to go in .0000… seconds. Lo, you have reached point B.
That also answers this:
I have that superpower, and thus in between typing the D and the O of the word “done”, I have counted them all.
To save Erik the trouble - so what was the value of the last one?
That is the point of his task- not that you can’t do it - but that you can (and in his view - must) then be able to identify the “last” value.
You are trying to use mathematical terms of art as if they meant what the dictionary says they do, you poor bastard. 
‘Countable’ means “You can take every element of this collection and label it with an integer without missing any.” That sequence is, therefore, countable. So is the sequence of nines in 0.999…
Frankly, ‘listable’ is a much better term for the concept, because what we really mean when we say the reals are ‘uncountable’ is that “If you tried to make a list of them, even an infinite one, you’re guaranteed to leave some out.” That’s the heart of Cantor’s diagonal argument.
Or, for those who get nervous about infinite lists, “No matter what method you come up with for assigning integers to reals, there will always be an infinite number of reals that no integer ends up assigned to.”
It occurs to me that part of Erik’s reasoning depends upon an ability to be sure of every value of the infinite expansion, which is a very limited case of converging infinite series.
So, rather than pick 1 as the target value, why not pick a transcendental? There are many infinite series that converge to transcendental. We can pick any one of them, perhaps restricting ourselves to those with all positive terms to avoid any oddities.
This one is useful:
Pi[sup]2[/sup]/6 = 1/1[sup]2[/sup] + 1/2[sup]2[/sup] + 1/3[sup]2[/sup] + 1/4[sup]2[/sup] + …
So, I walk along and cover the distance to Pi[sup]2[/sup]/6. I have covered all of the terms of the infinite sequence, and reach it. So clearly by Erik’s reasoning I have the last value of the infinite sequence. I wonder what it is?
Another sequence that is interesting is one of the sequences for Pi itself.
Pi = 3 + 4/(234) - 4/(456) + 4/(678) - 4/(8910) …
This rattles around Pi from above and below, so we don’t cover all values just walking from 0 to Pi. Which raise another interesting question. If I walk from 0 to Pi, I must have covered half of the values in the sequence when reaching Pi. If I can find the “last” value in the sequence this way, I must surely be able to find the “last” value coming from below, and the “last” value if I approach from above. Having both these values I must surely be able to tell if it the “last” value of the total series is greater or less than Pi.
Only if the list must be *countably *infinite.
In that context, “list” meant “series of entries each numbered by an integer”. Which was pretty well implied by context.
My point, though, was twofold: One, you can’t try to understand terms of art using dictionary definitions. Two, mathematicians sometimes pick really lousy words to use as terms of art.
In which case “list” looks the same as your definition of “countable”, except with some added complexity. (I don’t see why “list” is meant to be better or more explanatory than “countable”. And I took it the “what we really mean” comment was to imply some contrast between the two.)
Or reuse them (e.g. “consistency”).
I should be using “listable” here.
RE: “Correct. So why bother to select some particular set of points?”
Me: It is simply a reference point. One could have selected any number of different infinite series. The paradox of motion is one that relates an infinite series to the actual physical world. I intentionally wanted to select such a real world example that most would be familiar with to make the proof more “accessible”. And to address your comment on how airplanes fly below, yes it does matter if one is talking about a real physical world problem, which the paradox of motion is.
My goal in my proof is to construct the number 1.000… and 0.999… such that they are more than just some theoretical endless series of numbers or mathematical definition. This is the point of me starting out with Zeno’s Paradox, and I do not state it as a real Paradox, but rather right in the proof I state the solution. It is the solution that is important. And my solution (not literally “mine” but the one I choose as there are several alternatives) is the one that goes as follows:
As we progress through the infinite set of halves, (which I think we have all agreed to be countable, but perhaps they are not?), we have two corresponding series:
Time(hours): 1/2 + 1/4 + 1/8 + … = x
Distance(miles): 1/2 + 1/4 + 1/8 + … = x
I am purposefully leaving the answer as “x” because without limits we have no definite way to answer this. Again I am leaving limits out because they are not crucial to solving the paradox of motion. What is crucial is simple that there is an infinite set here which has a real world existence.
Furthermore I am asking that we map 1-1-1 the following:
Distance -> 1.000… -> 0.999…
Note that I don’t include time as it’s the same and only relevant as part of the solution to the paradox of motion. Also not to be more precise I ask that we count our starting point in movement from point A to B as “1.” And “0.” In the series above so that the actual mapping is like this:
Distance traveled:{0, 1/2, 3/4, 7/8, …}
1.000…: {1., 0, 0, 0, …}
0.999…: {0., 9, 9, 9, …}
Let me add to why I am leaving limits out of this. Without limits one cannot say with any proof what the final sum is in the distance traveled series. Maybe it’s 1 as the limit says, or maybe it’s .999.. as I propose. And since that is somewhat the argument at hand, I don’t want to confuse the issue. All I ask is that we accept the Distance traveled series as infinite and that we may map as I have describe this to 1.000… and 0.999…? Fair enough?
I can see some people pointing out that if we take the distance traveled series as summing to .999… then we have not solved the paradox of motion. To which I would reply, no at the end of that series we have not reached point B. It requires one further step that of distance 1/infinity miles which will take us 1/infinity hours. I don’t think this is relevant, but we can discuss if desired.
What I wish to show is that we have in principle constructed in their entirety the numbers 1.000… and 0.999… and not just in some fantasy world, but it has a real world correspondence.
Having these two numbers at hand it is an easy matter to see by ordinary arithmetic that their difference is:
0.000…1 or 1/infinity
The ability to this “ordinary” subtraction does rely on the fact that we indeed have our hands on metaphorically speaking the final member of these two infinite decimals. Having our hands on may be too string an implication, but rather that they must exist and we know what their values is may are.
Now I think this is where Frylock, (and others), and I are having some difference of opinion… that is that there is a final member. So I will leave this here and pick that up with my response to Frylock.
Also RE: “…There are a different number of points, because one is a countable set, and the other uncountable…”
How do come to this conclusion? I see no reason to believe that is the definition of an infnite decimal exmpansion as the uncountable kind? The set of reals is uncountable, but the decimal expansion? What is you source on that?
Indeed I did state some posts ago, that if we were to define .999… as the uncountable type of infnity then I may be incined to agree that .999… = 1. Maybe
Re: “But at some point after the interval within which I was doing the counting, I will find that I am not counting any more”
Hmm, I find that a bit of non-answer. I would think at least that “point after the interval” would be immediate, and assuming I also gave you the ability to write these down, you could show me the answer in writing. The scenario may be strictly impossible, fair enough. But moving from point A to point B is not. And is none the less an infinite set of halfway points.
I guess to me we have brought to surface a deeper paradox within the paradox of motion, which although seems quite adequately solved by assuming each successive halfway point takes half the time thus by the end we are usurping an infinite number of halfway points / unit of time. But in my mind still leaves open the question of how does one finish, complete, come to the end of, surpass even an infinite set without some final member?
It would seem that you have no trouble with this. As you have put it I think, (I’m sure you will correct me if I am wrong), there is no final member but still nothing that prevents us from iterating through all of them? We can pass through all the points, but in the end there was no end? I must say I envy your deeper understanding of traversing infinite sets than I have. Is there any way you could explain this in more detail?
I mean I agree that a final member of an infinite set seems contradictory. As in the impossible task I set out for you, if you were to name the last number before you stopped having to count as you say, I would say well what about that number + 1? And yet I just don’t see how you could ever be done without naming some last number? It is easy to say well we can never do such a thing to begin with, but the set of infinite halfway points in our apparent paradox of motion I would argue has the same properties of the set of all natural numbers. You would like to say we can pass through them all no problem, as we all do on a daily basis in moving from point A to point B, but that there is no last member?
I hope don’t mind if i use your likeness in my little script here. All names and statements are strictly fictional and reflect only the authors imagination.
Me: Hey Saltire, I was told .999… = 1? Is that true?
Saltire: Yes it is, Erik150x.
Me: How do you know?
Saltire: The Limit of .999… tells us so.
Me: How so?
Satire: Well .999… is just the sum of an infinite series defined as:
Sigma (x=1 to infinity) 9/(10^x)
and the limit of this series is 1
Me: okay, what is the Limit of 1/x as x-> infinity?
Saltire: it’s 0.
Me: okay, so 1/infinity = 0?
Saltire: no.
Me: So the Limit of something is not always the same as that something?
Saltire: That’s correct.
Me: How do you know when it gives you the correct answer and when it does not?
Saltire: Well, I suppose you don’t “know” in any absolute sense. There is no proof that shows in general the Limit of a series or function gives you “the” correct answer. In a sense Limits give us the answer we already expect should be the answer, but cannot be directly answered with ordinary math. One could view Limits as not giving you “the” answer, but rather the “limit” of the answer. Take for example 1/x as x->infinity. The limit does NOT tells us what 1/infinity IS, but rather exactly as stated what 1/x is approaching/converging to as x approaches infinity.
Me: Very interesting, Saltire. So in the case of Sigma [9/10^x] as x-> infinity it does not tell what it is at infinity, but rather what it is converging to as x->infinity?
Saltire: That is correct.
Me: Does the series not also converge to .999… as well as 1.000… ?
Saltire: It does indeed, but 1 - .999… = 0 and thus they must be the same number. We cannot have two real numbers with a difference of Zero. It would muck up the works sort of speak.
Me: But doesn’t the statement 1 - .999… = 0 assume they are equal. Why could I not say the difference is .000…1 or 1/infinity.
Saltire: Well those are not “numbers” defined in the Real number system. They simply don’t exist.
Me: Why not?
Saltire: Probably no single answer to that. One, we simply don’t need them in the overwhelmingly vast majority of real world problems. Two, at the time of defining Real numbers there was no consistent way to deal with 1/infinity or what is also called infinitesimals. Three, some people think the whole concept of an infinitely small number is utter non-sense, so they would prefer not admit their existence.
Me: Those seem like good answers, except for the third one, I don’t like that one at all.
Saltire: Well you’re not alone there, Erik150x. In fact there are more modern number systems which extend the real numbers to include the infinitely large as well as infinitely small.
They’re referred to as Hyperreals. In fact in the 1960’s, Abraham Robinson proved that the Hyperreals were logically consistent if and only if the reals were. And this put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated.
Me: That’s very interesting as well. To me the question of does .999… = 1 reflects a deeper question that of how one defines these numbers within the Real number system. But rather can one number be infinitely close to another. Putting Limits and the definition of real numbers aside, the question as I have stated it a number of times on here really reflect our very intuitive and reasonable notion that: 9/10 + 9/100 + 9/1000 + … does not ever actually equal one but becomes infinitely close to 1. Taking the Limit of this series is a bit circular in the sense that by the very definition of a Limit, there two numbers .999… and 1 are equal. It does not really address the intuitive notion that some people have which believes 2 numbers can be infinitely close to one another.
Saltire: Understandable position I think, Erik150x. But you have also not addressed mine in which I point out that to accept Hyperreals, you would have to accepts notions like an infinitesimal number x can have the property that x^2 = 0? Or at least if you can accept that, why can you not accept that .999… = 1?
Me: Well, I don’t think that I have ever said on here I could never accept it, only that I don’t believe it is and that there is no proof which extends from our most basic understanding of numbers and ordinary arithmetic. Again the real number system and limits system ask us to accept this notion, which while maybe for some should be a given, I do not think for all of us it is a given notion. After all how does .999… ever become 1? The most intuitive notion would be that it does not. If it does, I would like to know how. I think the best one could say is that as the series reaches an infinite number of decimal places “something” just happens to eliminate any difference between the two. I am not satisfied with this. I am no expert what so ever on the philosophy of infinitesimals. But people seem to have no trouble with infinity + 1 = infinity and such matters, I would expect infinitesimals would also have some properties which do not follow those of ordinary arithmetic. Lastly I do agree 3/3=1, but have no idea why you ask. I suppose you will expand on that?
You handwaved away this proof when it was offered before, but here it is again.
10 * 0.9… = 9.9…
10 * 0.9… - 0.9… = 9.9… - 0.9…
(10-1) * 0.9… = 9.9… - 0.9…
(10-1) * 0.9… = 9
9 * 0.9… = 9
0.9… = 1
It’s hard to see how you could argue with any of the steps in this proof except for the first one.
So … if 10*0.9… is not equal to 9.9…, then what is it equal to?
Because
1/3 = 0.333…
1/3+1/3+1/3 = 0.333… + 0.333… + 0.333… = 0.999… = 1
perhaps?
erik140x, you have failed to convince us. Even if you did convince us, what do you think we could do? Do you think that we’re the secret council of mathematicians who control the world of mathematical definitions? We control nothing. If you want to convince the world of your theories, get a Ph.D. in math and become a big-name mathematician. Publish your theories. Maybe then you could convince the mathematical community. Or if you want to bypass the mathematical community, publish your theories in ordinary books. Maybe you can convince the people of the world of your theories and ordinary people will unite to kill all mathematicians because they refuse to accept your theories.
I have net access for a bit. Hurray!
First, zombywoof has stated exactly why I mentioned 3/3=1. I don’t think you were really unclear on that, since it’s been mentioned in this thread before, and it should be clear to an intelligent and honest student of mathematics.
I just skimmed the wiki article on hyperreals, and found a mention that division by zero is not allowed in hyperreal math, just as with reals. I didn’t find it stated explicitly, but I think I can assume that division by infinity is also not defined. However, I don’t really understand hyperreals, and I don’t think they need to be in this discussion.
In your hypothetical conversation, you have me saying a lot of stuff about limits, and what 1/∞ equals. Note that limits were not part of the post you are replying to, and that in that post, I stated that 1/∞ is not defined in the math I’m discussing. So, needless to say, most of what you typed was useless as a reply to my post.
You have insisted that someone solve this problem without limits (which has been done, though you ignore it). Let me ask you to state your case without using infinity in arithmetic as if it were a number.
Let me say it again: 0.999… = 1 is not about limits, it’s an artifact of decimal notation. State the same equation in a different base, or in fractions [nine ninths is one whole], and the problem doesn’t exist. Please stop pretending you don’t know this. Or do you have some reason to state that the results of calculations converted to other notations don’t apply in base-10 with decimal fractions?
Your proof has been discussed and refuted many times in this thread.
Let’s look at 10 * .9
10 * .9 = 9.0 not 9.9
now you can say 10 * .999… = 9.999… in our real number system ONLY because we define our real numbers in terms of limits. What you are doing there is throwing in an extra .000…9 at the end or 9/infinity. It’s kind of sad that people throw that proof out there without understanding what’s really going on there. Not everyone, mind you, but I would say most. There is really no point in the proof other than to deceive one into thinking the .000…1 difference in 1 vs .999… just disappears. You might as well just offer up the statement that the real number system defines .999… as equal to one. The is no need to prove anything, as the real number system nor limits does anything of the sort.
“To summarize: The real question is not why 0.9999…=1, but rather, what on earth are real numbers in the first place? There is one tempting answer, according to which, 0.9999… is actually not equal to 1. But this answer is nonstandard because it creates real numbers with very badly behaved subtraction. To avoid the problems with bad subtraction, mathematicians basically throw up their hands and declare that 0.9999…=1, though they cloak this with fancy-sounding words like “equivalence class” and “equivalence relation” (or they avoid it altogether using more sophisticated methods of defining the reals).” - http://www.xamuel.com/why-is-0point999-1/
Do the math by hand. You will see as I am sure you once did doing the division long hand, that there is always a remainder:
1/3 = .333… + 1/[3*10^infinity]
Thus 3 * 1/3 = 3 * (.333… + 1/[3*10^infinity]) = .999… + 1/10^infnity